On Smooth Mesoscopic Linear Statistics of the Eigenvalues of Random Permutation Matrices

We study the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group. If we apply a smooth enough test function f to all the determinations of the eigenangles of the permutations, we get a convergence in distribution when the order of the permutation tends to infinity. Two distinct kinds of limit appear: if $$f(0)\ne 0$$ f ( 0 ) ≠ 0 , we have a central limit theorem with a logarithmic variance; and if $$f(0) = 0$$ f ( 0 ) = 0 , the convergence holds without normalization and the limit involves a scale-invariant Poisson point process.

Effective bounds of the variance of statistics on multisets of necklaces

The variance of a linear statistics on multisets of necklaces is explored. The upper and lower bounds with optimal constants are obtained

Pair dependent linear statistics for CβE

We study the limiting distribution of a pair counting statistics of the form [Formula: see text] for the circular [Formula: see text]-ensemble (C[Formula: see text]E) of random matrices for sufficiently smooth test function [Formula: see text] and [Formula: see text] For [Formula: see text] and [Formula: see text] our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.

Spectrum of SYK model II: Central limit theorem

In our previous paper [R. Feng, G. Tian and D. Wei, Spectrum of SYK model, Peking Math. J. 2 (2019) 41–70], we derived the almost sure convergence of the global density of eigenvalues of random matrices of the SYK model. In this paper, we will prove the central limit theorem for the linear statistics of eigenvalues of the SYK model and compute its variance.

An Assumption-Free Exact Test For Fixed-Design Linear Models With Exchangeable Errors

We propose the cyclic permutation test to test general linear hypotheses for linear models. This test is nonrandomized and valid in finite samples with exact Type-I error α for an arbitrary fixed design matrix and arbitrary exchangeable errors, whenever 1 / α is an integer and n / p ≥ 1 / α – 1. The test applies the marginal rank test on 1 / α linear statistics of the outcome vector where the coefficient vectors are determined by solving a linear system such that the joint distribution of the linear statistics is invariant to a nonstandard cyclic permutation group under the null hypothesis. The power can be further enhanced by solving a secondary nonlinear travelling salesman problem, for which the genetic algorithm can find a reasonably good solution. We show that the Cyclic Permutation Test has comparable power with existing tests through extensive simulation studies. When testing for a single contrast of coefficients, an exact confidence interval can be obtained by inverting the test. Furthermore, we provide a selective yet extensive literature review of the century-long efforts on this problem, highlighting the novelty of our test.